Chapter2_Notes

Of applied electromagnetics ulaby et al for ece331

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Unformatted text preview: 9 • More interesting than the instantaneous power is the time-average power, which can be obtained from Pav = 1 T T P (t)dt = 0 ω 2π 2π/ω P (t)dt (155) 0 • We can use the identity, cos2 x = 1 (1 + cos 2x) 2 (156) And, remember that the integral of cos ωt over a period is 0, that is, powers, yields, T cos ωtdt = 0 (157) 0 • Expanding the incident power out, P i (0, t) = |V0+ |2 (1/2 + 1/2 cos(ωt + φ+ )) (W) Z0 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (158) Electromagnetics I: Transmission lines so we have i Pav = 80 T 1 T P (t)dt = 0 |V0+ |2 2Z0 (159) • The reflected power is found similarly, so we have the incident and reflected average power (in Watts), |V0+ |2 (W) 2Z0 (160) |V0+ |2 i = −|Γ|2 Pav 2Z0 (161) i Pav = r Pav = −|Γ|2 − r Pav = |Γ|2 i Pav (162) • This is an important result: the ratio of the reflected and incident powers at the load give the reflection coefficient magnitude squared, or that the reflected power at the load is equal to the incident power reduced by |Γ|2 term. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Electromagnetics I: Transmission lines 81 • The net average power delivered to the load is a sum of the incident and reflected powers: i r Pav = Pav + Pav = |V0+ |2 1 − |Γ|2 (W) 2Z0 (163) • Note, that we could have done the same thing (even easier) using a phasor representation. • The starting point for that is the average power relationship which is, 1 ˜˜ V · I∗ (164) Pav = 2 • But, starting with that we still end up with, Pav = |V0+ |2 1 − |Γ|2 (W) 2Z0 Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (165) Electromagnetics I: Transmission lines Zg 82 Transmission line i Pav + ~ Vg r i Pav = |Γ|2 Pav - ZL Figure 20: Power flows on a transmission line. Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. Figure 2-19 Electromagnetics I: Transmission lines 83 1.10. Smith Chart The Smith chart is a graphical tool for representing impedances, reflection coefficients and transmission lines. Let’s start with the polar representation of the reflection coefficient, Γ. Γ = |Γ|ejθr = |Γ| cos θr + j |Γ| sin θr = Γr + j Γi (166) • For passive load impedances, the reflection coefficient is less then unity, i.e. |Γ| ≤ 1. • Impedances on the Smith chart are normalized (i.e. divided) by a normalizing impedance. In our cases, that will be characteristic impedance Z0 of a transmission line, so that zL = ZL /Z0 = rL + jxL . • In that case we have, Γ= 1+Γ zL − 1 , or zL = zL + 1 1−Γ Notes based on Fundamentals of Applied Electromagnetics (Ulaby et al) for ECE331, PSU. (167) Electromagnetics I: Transmission lines 84 Gi 1 ° 2 20 qr = |G | = 1 D -1 -0.9 -0.7 -0.5 -0.3 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -0.1 3 =5 qr q r = 180° Short-circuit load q r = 90° A |G ° A| = 0.5 C 0.1 0.3 B -0.2 |G B| = 0.54 -0.3 -0.4 -0.5 -0.6 -...
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